Finite size effect of ions and dipoles close to charged interfaces

The modified dipolar Poisson--Boltzmann (MDPB) equation, where the electrostatics of the dipolar interactions of solvent molecules and also the finite size effects of ions and dipolar solvent molecules are explicitly taken into account on a mean-field level, is studied numerically for a two-plate system with oppositely charged surfaces. The MDPB equation is solved numerically, using the nonlinear Multigrid method, for one-dimensional finite volume meshes. For a high enough surface charge density, numerical results of the MDPB equation reveal that the effective dielectric constant decreases with the increase of the surface charge density. Furthermore, increasing the salt concentration leads to the decrease of the effective dielectric constant close to the charged surfaces. This decrease of the effective dielectric constant with the surface charge density is opposite to the trend from the dipolar Poisson--Boltzmann (DPB) equation. This seemingly inconsistent result is due to the fact that the mean-field approach breaks down in such highly charged systems where the counterions and dipoles are strongly attracted to the charged surfaces and form a quasi two-dimensional layer. In the weak-coupling regime with the electrostatic coupling parameter (the ratio of Bjerrum length to Gouy--Chapman length) $\varXi < 1$, where the MDPB equation works, the effective dielectric constant is independent of the distance from the charged surfaces and there is no accumulation of dipoles near the charged surfaces. Therefore, there are no physical and computational advantages for the MDPB equation over the modified Poisson--Boltzmann (MPB) equation where the effect of dipolar interactions of solvent dipoles is implicitly taken into account in the renormalised dielectric constant.